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Chapter 1
The Trigonometric Functions
Trigonometry had its start as the study of triangles. The study of triangles can be
traced back to the second millenium B.C.E. in Egyptian and Babylonian mathemat-
ics. In fact, the word trigonmetry is derived from a Greek word meaning “triangle
measuring.” We will study the trigonometry of triangles in Chapter 3. Today,
however, the trigonometric functions are used in more ways. In particular, the
trigonometric functions can be used to model periodic phenomena such as sound
and light waves, the tides, the number of hours of daylight per day at a particular
location on earth, and many other phenomena that repeat values in specified time
intervals.
Our study of periodic phenomena will begin in Chapter 2, but first we must
study the trigonometric functions. To do so, we will use the basic form of a repeat-
ing (or periodic) phenomena of travelling around a circle at a constant rate.
1
2 Chapter 1. The Trigonometric Functions
1.1 The Unit Circle
Focus Questions
The following questions are meant to guide our study of the material in this
section. After studying this section, we should understand the concepts mo-
tivated by these questions and be able to write precise, coherent answers to
these questions.
� What is the unit circle and why is it important in trigonometry? What is
the equation for the unit circle?
� What is meant by “wrapping the number line around the unit circle?”
How is this used to identify real numbers as the lengths of arcs on the
unit circle?
� How do we associate an arc on the unit circle with a closed interval of
real numbers?
Beginning Activity
As has been indicated, one of the primary reasons we study the trigonometric func-
tions is to be able to model periodic phenomena mathematically. Before we begin
our mathematical study of periodic phenomena, here is a little “thought experi-
ment” to consider.
Imagine you are standing at a point on a circle and you begin walking around
the circle at a constant rate in the counterclockwise direction. Also assume that
it takes you four minutes to walk completely around the circle one time. Now
suppose you are at a point P on this circle at a particular time t .
� Describe your position on the circle 2 minutes after the time t .
� Describe your position on the circle 4 minutes after the time t .
� Describe your position on the circle 6 minutes after the time t .
� Describe your position on the circle 8 minutes after the time t .
The idea here is that your position on the circle repeats every 4 minutes. After 2
minutes, you are at a point diametrically opposed from the point you started. After
1.1. The Unit Circle 3
4 minutes, you are back at your starting point. In fact, you will be back at your
starting point after 8 minutes, 12 minutes, 16 minutes, and so on. This is the idea
of periodic behavior.
The Unit Circle and the Wrapping Function
In order to model periodic phenomena mathematically, we will need functions that
are themselves periodic. In other words, we look for functions whose values repeat
in regular and recognizable patterns. Familiar functions like polynomials and ex-
ponential functions don’t exhibit periodic behavior, so we turn to the trigonometric
functions. Before we can define these functions, however, we need a way to intro-
duce periodicity. We do so in a manner similar to the thought experiment, but we
also use mathematical objects and equations. The primary tool is something called
the wrapping function. Instead of using any circle, we will use the so-called unit
circle. This is the circle whose center is at the origin and whose radius is equal to
1, and the equation for the unit circle is x2 C y2 D 1.
Figure 1.1: Setting up to wrap the number line around the unit circle
Figure 1.1 shows the unit circle with a number line drawn tangent to the circle
at the point .1; 0/. We will “wrap” this number line around the unit circle. Unlike
the number line, the length once around the unit circle is finite. (Remember that
the formula for the circumference of a circle as 2�r where r is the radius, so the
length once around the unit circle is 2� .) However, we can still measure distances
and locate the points on the number line on the unit circle by wrapping the number
line around the circle. We wrap the positive part of this number line around the
circumference of the circle in a counterclockwise fashion and wrap the negative
4 Chapter 1. The Trigonometric Functions
part of the number line around the circumference of the unit circle in a clockwise
direction.
Two snapshots of an animation of this process for the counterclockwise wrap
are shown in Figure 1.2 and two such snapshots are shown in Figure 1.3 for the
clockwise wrap.
Figure 1.2: Wrapping the positive number line around the unit circle
Figure 1.3: Wrapping the negative number line around the unit circle
Following is a link to an actual animation of this process, including both posi-
tive wraps and negative wraps.
http://gvsu.edu/s/Kr
Figure 1.2 and Figure 1.3 only show a portion of the number line being wrapped
around the circle. Since the number line is infinitely long, it will wrap around
1.1. The Unit Circle 5
the circle infinitely many times. A result of this is that infinitely many different
numbers from the number line get wrapped to the same location on the unit circle.
� The number 0 and the numbers 2� , �2� , and 4� (as well as others) get
wrapped to the point .1; 0/. We will usually say that these points get mapped
to the point .1; 0/.
� The number�
2is mapped to the point .0; 1/. This is because the circum-
ference of the unit circle is 2� and so one-fourth of the circumference is1
4.2�/ D �
2.
� If we now add 2� to�
2, we see that
5�
2also gets mapped to .0; 1/. If we
subtract 2� from�
2, we see that �3�
2also gets mapped to .0; 1/.
However, the fact that infinitely many different numbers from the number line get
wrapped to the same location on the unit circle turns out to be very helpful as it
will allow us to model and represent behavior that repeats or is periodic in nature.
Progress Check 1.1 (The Unit Circle.)
1. Find two different numbers, one positive and one negative, from the number
line that get wrapped to the point .�1; 0/ on the unit circle.
2. Describe all of the numbers on the number line that get wrapped to the point
.�1; 0/ on the unit circle.
3. Find two different numbers, one positive and one negative, from the number
line that get wrapped to the point .0; 1/ on the unit circle.
4. Find two different numbers, one positive and one negative, from the number
line that get wrapped to the point .0;�1/ on the unit circle.
One thing we should see from our work in Progress Check 1.1 is that integer
multiples of � are wrapped either to the point .1; 0/ or .�1; 0/ and that odd inte-
ger multiples of�
2are wrapped to either to the point .0; 1/ or .0;�1/. Since the
circumference of the unit circle is 2� , it is not surprising that fractional parts of �
and the integer multiples of these fractional parts of � can be located on the unit
circle. This will be studied in the next progress check.
6 Chapter 1. The Trigonometric Functions
Progress Check 1.2 (The Unit Circle and �).
The following diagram is a unit circle with 24 equally spaced points plotted on the
circle. Since the circumference of the circle is 2� units, the increment between
two consecutive points on the circle is2�
24D �
12.
Label each point with the smallest nonnegative real number t to which it corre-
sponds. For example, the point .1; 0/ on the x-axis corresponds to t D 0. Moving
counterclockwise from this point, the second point corresponds to2�
12D �
6.
π
12
π
6
π
4
3
π
12
7
π
3
5
π
4
5
π
6
7
π
12
5
π
6
11
π
12
23
Figure 1.4: Points on the unit circle
Using Figure 1.4, approximate the x-coordinate and the y-coordinate of each
of the following:
1. The point on the unit circle that corresponds to t D �
3.
2. The point on the unit circle that corresponds to t D 2�
3.
3. The point on the unit circle that corresponds to t D 4�
3.
4. The point on the unit circle that corresponds to t D 5�
3.
1.1. The Unit Circle 7
5. The point on the unit circle that corresponds to t D �
4.
6. The point on the unit circle that corresponds to t D 7�
4.
Arcs on the Unit Circle
When we wrap the number line around the unit circle, any closed interval on the
number line gets mapped to a continuous piece of the unit circle. These pieces
are called arcs of the circle. For example, the segmenth
0;�
2
i
on the number line
gets mapped to the arc connecting the points .1; 0/ and .0; 1/ on the unit circle as
shown in Figure 1.5. In general, when a closed interval Œa; b� is mapped to an arc
on the unit circle, the point corresponding to t D a is called the initial point of
the arc, and the point corresponding to t D b is called the terminal point of the
arc. So the arc corresponding to the closed intervalh
0;�
2
i
has initial point .1; 0/
and terminal point .0; 1/.
arc
x
y
x2 + y2 = 1
Figure 1.5: An arc on the unit circle
Progress Check 1.3 (Arcs on the Unit Circle).
Draw the following arcs on the unit circle.
1. The arc that is determined by the intervalh
0;�
4
i
on the number line.
8 Chapter 1. The Trigonometric Functions
2. The arc that is determined by the interval
�
0;2�
3
�
on the number line.
3. The arc that is determined by the interval Œ0;��� on the number line.
Coordinates of Points on the Unit Circle
When we have an equation (usually in terms of x and y) for a curve in the plane and
we know one of the coordinates of a point on that curve, we can use the equation
to determine the other coordinate for the point on the curve. The equation for the
unit circle is x2 C y2 D 1. So if we know one of the two coordinates of a point
on the unit circle, we can substitute that value into the equation and solve for the
value(s) of the other variable.
For example, suppose we know that the x-coordinate of a point on the unit
circle is �1
3. This is illustrated on the following diagram. This diagram shows
the unit circle (x2 C y2 D 1) and the vertical line x D �1
3. This shows that
there are two points on the unit circle whose x-coordinate is �1
3. We can find
the y-coordinates by substituting the x-value into the equation and solving for y.
x
y
1x = -
3
_
x2 C y2 D 1�
�1
3
�2
C y2 D 1
1
9C y2 D 1
y2 D 8
9
Since y2 D 8
9, we see that y D ˙
r
8
9and so y D ˙
p8
3. So the two points
on the unit circle whose x-coordinate is �1
3are
�1
3;
p8
3
!
; which is in the second quadrant, and
�1
3;�p
8
3
!
; which is in the third quadrant.
1.1. The Unit Circle 9
The first point is in the second quadrant and the second point is in the third quad-
rant. We can now use a calculator to verify that
p8
3� 0:9428. This seems
consistent with the diagram we used for this problem.
Progress Check 1.4 (Points on the Unit Circle.)
1. Find all points on the unit circle whose y-coordinate is1
2.
2. Find all points on the unit circle whose x-coordinate is
p5
4.
Summary of Section 1.1
In this section, we studied the following important concepts and ideas:
� The unit circle is the circle of radius 1 that is centered at the origin. The
equation of the unit circle is x2 C y2 D 1. It is important because we will
use this as a tool to model periodic phenomena.
� We “wrap” the number line about the unit circle by drawing a number line
that is tangent to the unit circle at the point .1; 0/. We wrap the positive part
of the number line around the unit circle in the counterclockwise direction
and wrap the negative part of the number line around the unit circle in the
clockwise direction.
� When we wrap the number line around the unit circle, any closed interval of
real numbers gets mapped to a continuous piece of the unit circle, which is
called an arc of the circle. When the closed interval Œa; b� is mapped to an
arc on the unit circle, the point corresponding to t D a is called the initial
point of the arc, and the point corresponding to t D b is called the terminal
point of the arc.
Exercises for Section 1.1
1. The following diagram shows eight points plotted on the unit circle. These
points correspond to the following values when the number line is wrapped
around the unit circle.
t D 1; t D 2; t D 3; t D 4; t D 5; t D 6; t D 7; and t D 9:
10 Chapter 1. The Trigonometric Functions
x
y
(a) Label each point in the diagram with its value of t .
? (b) Approximate the coordinates of the points corresponding to t D 1,
t D 5, and t D 9.
? 2. The following diagram shows the points corresponding to t D �
5and t D
2�
5when the number line is wrapped around the unit circle.
x
y
On this unit circle, draw the points corresponding to t D 4�
5, t D 6�
5,
t D 8�
5, and t D 10�
5.
3. Draw the following arcs on the unit circle.
1.1. The Unit Circle 11
(a) The arc that is determined by the intervalh
0;�
6
i
on the number line.
(b) The arc that is determined by the interval
�
0;7�
6
�
on the number line.
(c) The arc that is determined by the interval Œ0;��
3� on the number line.
(d) The arc that is determined by the interval Œ0;�4�
5� on the number line.
? 4. Determine the quadrant that contains the terminal point of each given arc
with initial point .1; 0/ on the unit circle.
(a)7�
4
(b) �7�
4
(c)3�
5
(d)�3�
5
(e)7�
3
(f)�7�
3
(g)5�
8
(h)�5�
8(i) 2:5
(j) �2:5
(k) 3
(l) 3C 2�
(m) 3 � �
(n) 3 � 2�
5. Find all the points on the unit circle:
? (a) Whose x-coordinate is1
3.
? (b) Whose y-coordinate is �1
2.
(c) Whose x-coordinate is �3
5.
(d) Whose y-coordinate is �3
4and whose x-coordinate is negative.
12 Chapter 1. The Trigonometric Functions
1.2 The Cosine and Sine Functions
Focus Questions
The following questions are meant to guide our study of the material in this
section. After studying this section, we should understand the concepts mo-
tivated by these questions and be able to write precise, coherent answers to
these questions.
� If the real number t represents the (signed) length of an arc, how do we
define cos.t/ and sin.t/?
� In what quadrants (of the terminal point of an arc t on the unit circle) is
cos.t/ positive (negative)? In what quadrants (of the terminal point of
an arc t on the unit circle) is sin.t/ positive (negative)?
� What is the Pythagorean Identity? How is this identity derived from the
equation for the unit circle?
Beginning Activity
1. What is the unit circle? What is the equation of the unit circle?
2. Review Progress Check 1.4 on page 9.
3. Review the completed version of Figure 1.4 that is in the answers for Progress
Check 1.2 on page 6.
4. (a) What is the terminal point of the arc on the unit circle that corresponds
to the intervalh
0;�
2
i
?
(b) What is the terminal point of the arc on the unit circle that corresponds
to the interval Œ0; ��?
(c) What is the terminal point of the arc on the unit circle that corresponds
to the interval
�
0;3�
2
�
?
(d) What is the terminal point of the arc on the unit circle that corresponds
to the intervalh
0;��
2
i
?
1.2. The Cosine and Sine Functions 13
The Cosine and Sine Functions
We started our study of trigonometry by learning about the unit circle, how to
wrap the number line around the unit circle, and how to construct arcs on the unit
circle. We are now able to use these ideas to define the two major circular, or
trigonometric, functions. These circular functions will allow us to model periodic
phenomena such as tides, the amount of sunlight during the days of the year, orbits
of planets, and many others.
It may seem like the unit circle is a fairly simple object and of little interest, but
mathematicians can almost always find something fascinating in even such simple
objects. For example, we define the two major circular functions, the cosine and
sine1 in terms of the unit circle as follows. Figure 1.6 shows an arc of length t on
the unit circle. This arc begins at the point .1; 0/ and ends at its terminal point P.t/.
We then define the cosine and sine of the arc t as the x and y coordinates of the
point P , so that P.t/ D .cos.t/; sin.t// (we abbreviate the cosine as cos and the
sine as sin). So the cosine and sine values are determined by the arc t and the cosine
x
y
t
P(t) = (cos(t), sin(t))
2x 2y+ = 1
Figure 1.6: The Circular Functions
1According to the web site Earliest Known Uses of Some of the Words of Mathematics at
http://jeff560.tripod.com/mathword.html , the origin of the word sine is Sanskrit
through Arabic and Latin. While the accounts of the actual origin differ, it appears that the Sanskrit
work “jya” (chord) was taken into Arabic as “jiba”, but was then translated into Latin as “jaib” (bay)
which became “sinus” (bay or curve). This word was then anglicized to become our “sine”. The
word cosine began with Plato of Tivoli who use the expression “chorda residui”. While the Latin
word chorda was a better translation of the Sanskrit-Arabic word for sine than the word sinus, that
word was already in use. Thus, “chorda residui” became “cosine”.
14 Chapter 1. The Trigonometric Functions
and sine are functions of the arc t . Since the arc lies on the unit circle, we call the
cosine and sine circular functions. An important part of trigonometry is the study
of the cosine and sine and the periodic phenomena that these functions can model.
This is one reason why the circular functions are also called the trigonometric
functions.
Note: In mathematics, we always create formal definitions for objects we com-
monly use. Definitions are critically important because with agreed upon defini-
tions, everyone will have a common understanding of what the terms mean. With-
out such a common understanding, there would be a great deal of confusion since
different people would have different meanings for various terms. So careful and
precise definitions are necessary in order to develop mathematical properties of
these objects. In order to learn and understand trigonometry, a person needs to be
able to explain how the circular functions are defined. So now is a good time to
start working on understanding these definitions.
Definition. If the real number t is the directed length of an arc (either positive
or negative) measured on the unit circle x2 C y2 D 1 (with counterclockwise
as the positive direction) with initial point (1, 0) and terminal point .x; y/,
then the cosine of t , denoted cos.t/, and sine of t , denoted sin.t/, are defined
to be
cos.t/ D x and sin.t/ D y:
Figure 1.6 illustrates these definitions for an arc whose terminal point is in the
first quadrant.
At this time, it is not possible to determine the exact values of the cosine and
sine functions for specific values of t . It can be done, however, if the terminal
point of an arc of length t lies on the x-axis or the y-axis. For example, since the
circumference of the unit circle is 2� , an arc of length t D � will have it terminal
point half-way around the circle from the point .1; 0/. That is, the terminal point is
at .�1; 0/. Therefore,
cos.�/ D �1 and sin.�/ D 0:
Progress Check 1.5 (Cosine and Sine Values).
Determine the exact values of each of the following:
1.2. The Cosine and Sine Functions 15
1. cos��
2
�
and sin��
2
�
.
2. cos
�
3�
2
�
and sin
�
3�
2
�
.
3. cos .0/ and sin .0/.
4. cos�
��
2
�
and sin�
��
2
�
.
5. cos .2�/ and sin .2�/.
6. cos .��/ and sin .��/.
Important Note: Since the cosine and sine are functions of an arc whose length is
the real number t , the input t determines the output of the cosine and sine functions.
As a result, it is necessary to specify the input value when working with the cosine
and sine. In other words, we ALWAYS write cos.t/ where t is the real number
input, and NEVER just cos. To reiterate, the cosine and sine are functions, so we
MUST indicate the input to these functions.
Progress Check 1.6 (Approximating Cosine and Sine Values).
For this progress check, we will use the Geogebra Applet called Terminal Points of
Arcs on the Unit Circle. A web address for this applet is
http://gvsu.edu/s/JY
For this applet, we control the value of the input t with the slider for t . The values
of t range from �20 to 20 in increments of 0:5. For a given value of t , an arc is
drawn of length t and the approximate coordinates of the terminal point of that arc
are displayed. Use this applet to find approximate values for each of the following:
1. cos.1/ and sin.1/.
2. cos.2/ and sin.2/.
3. cos.�4/ and sin.�4/.
4. cos.5:5/ and sin.5:5/.
5. cos.15/ and sin.15/.
6. cos.�15/ and sin.�15/.
Some Properties of the Cosine and Sine Functions
The cosine and sine functions are called circular functions because their values
are determined by the coordinates of points on the unit circle. For each real number
t , there is a corresponding arc starting at the point .1; 0/ of (directed) length t that
lies on the unit circle. The coordinates of the end point of this arc determines the
values of cos.t/ and sin.t/.
In previous mathematics courses, we have learned that the domain of a function
is the set of all inputs that give a defined output. We have also learned that the range
of a function is the set of all possible outputs of the function.
16 Chapter 1. The Trigonometric Functions
Progress Check 1.7 (Domain and Range of the Circular Functions.)
1. What is the domain of the cosine function? Why?
2. What is the domain of the sine function? Why?
3. What is the largest x coordinate that a point on the unit circle can have?
What is the smallest x coordinate that a point on the unit circle can have?
What does this tell us about the range of the cosine function? Why?
4. What is the largest y coordinate that a point on the unit circle can have?
What is the smallest y coordinate that a point on the unit circle can have?
What does this tell us about the range of the sine function? Why?
Although we may not be able to calculate the exact values for many inputs for
the cosine and sine functions, we can use our knowledge of the coordinate system
and its quadrants to determine if certain values of cosine and sine are positive or
negative. The idea is that the signs of the coordinates of a point P.x; y/ that is
plotted in the coordinate plane are determined by the quadrant in which the point
lies. (Unless it lies on one of the axes.) Figure 1.7 summarizes these results for the
signs of the cosine and sine function values. The left column in the table is for the
location of the terminal point of an arc determined by the real number t .
Quadrant cos.t/ sin.t/
QI positive positive
QII negative positive
QIII negative negative
QIV positive negative
x
y
sin (t) > 0 sin (t) > 0
cos (t) > 0
cos (t) > 0cos (t) < 0
cos (t) < 0
sin (t) < 0 sin (t) < 0
Figure 1.7: Signs of the cosine and sine functions
What we need to do now is to determine in which quadrant the terminal point
of an arc determined by a real number t lies. We can do this by once again using
the fact that the circumference of the unit circle is 2� , and when we move around
the unit circle from the point .1; 0/ in the positive (counterclockwise) direction, we
1.2. The Cosine and Sine Functions 17
will intersect one of the coordinate axes every quarter revolution. For example, if
0 < t <�
2, the terminal point of the arc determined by t is in the first quadrant
and cos.t/ > 0 and sin.t/ > 0.
Progress Check 1.8 (Signs of cos.t/ and sin.t/.)
1. If�
2< t < � , then what are the signs of cos.t/ and sin.t/?
2. If � < t <3�
2, then what are the signs of cos.t/ and sin.t/?
3. If3�
2< t < 2� , then what are the signs of cos.t/ and sin.t/?
4. If5�
2< t < 3� , then what are the signs of cos.t/ and sin.t/?
5. For which values of t (between 0 and 2�) is cos.t/ positive? Why?
6. For which values of t (between 0 and 2�) is sin.t/ positive? Why?
7. For which values of t (between 0 and 2�) is cos.t/ negative? Why?
8. For which values of t (between 0 and 2�) is sin.t/ negative? Why?
Progress Check 1.9 (Signs of cos.t/ and sin.t/ (Part 2))
Use the results summarized in Figure 1.7 to help determine if the following quan-
tities are positive, negative, or zero. (Do not use a calculator.)
1. cos��
5
�
2. sin��
5
�
3. cos
�
5�
8
�
4. sin
�
5�
8
�
5. cos
��9�
16
�
6. sin
��9�
16
�
7. cos
��25�
12
�
8. sin
��25�
12
�
18 Chapter 1. The Trigonometric Functions
The Pythagorean Identity
In mathematics, an identity is a statement that is true for all values of the vari-
ables for which it is defined. In previous courses, we have worked with algebraic
identities such as
7x C 12x D 19x a C b D b C a
a2 � b2 D .a C b/.a � b/ x.y C z/ D xy C xz
where it is understood that all the variables represent real numbers. In trigonome-
try, we will develop many so-called trigonometric identities. The following progress
check introduces one such identity between the cosine and sine functions.
Progress Check 1.10 (Introduction to the Pythagorean Identity)
We know that the equation for the unit circle is x2C y2 D 1. We also know that if
t is an real number, then the terminal point of the arc determined by t is the point
.cos.t/; sin.t// and that this point lies on the unit circle. Use this information to
develop an identity involving cos.t/ and sin.t/.
Using the definitions x D cos.t/ and y D sin.t/ along with the equation for
the unit circle, we obtain the following identity, which is perhaps the most
important trigonometric identity.
For each real number t; ..cos.t//2 C .sin.t//2 D 1:
This is called the Pythagorean Identity. We often use the shorthand nota-
tion cos2.t/ for .cos.t//2 and sin2.t/ for .sin.t//2 and write
For each real number t; cos2.t/C sin2.t/ D 1:
Important Note about Notation. Always remember that by cos2.t/ we mean
.cos.t//2. In addition, note that cos2.t/ is different from cos.t2/.
The Pythagorean Identity allows us to determine the value of cos.t/ or sin.t/ if
we know the value of the other one and the quadrant in which the terminal point of
arc t lies. This is illustrated in the next example.
Example 1.11 (Using the Pythagorean Identity)
Assume that cos.t/ D 2
5and the terminal point of arc .t/ lies in the fourth quadrant.
We will use this information to determine the value of sin.t/. The primary tool we
will use is the Pythagorean Identity, but please keep in mind that the terminal point
1.2. The Cosine and Sine Functions 19
for the arc t is the point .cos.t/; sin.t//. That is, x D cos.t/ and y D sin.t/. So
this problem is very similar to using the equation x2 C y2 D 1 for the unit circle
and substituting x D 2
5.
Using the Pythagorean Identity, we then see that
cos2.t/C sin2.t/ D 1�
2
5
�2
C sin2.t/ D 1
4
25C sin2.t/ D 1
sin2.t/ D 1 � 4
25
sin2.t/ D 21
25
This means that sin.t/ D ˙r
21
25, and since the terminal point of arc .t/ is in the
fourth quadrant, we know that sin.t/ < 0. Therefore, sin.t/ D �r
21
25. Since
p25 D 5, we can write
sin.t/ D �p
21p25D �p
21
5:
Progress Check 1.12 (Using the Pythagorean Identity)
1. If cos.t/ D 1
2and the terminal point of the arc t is in the fourth quadrant,
determine the value of sin.t/.
2. If sin.t/ D �2
3and � < t <
3�
2, determine the value of cos.t/.
20 Chapter 1. The Trigonometric Functions
Summary of Section 1.2
In this section, we studied the following important concepts and ideas:
� If the real number t is the directed
length of an arc (either positive or
negative) measured on the unit circle
x2 C y2 D 1 (with counterclockwise
as the positive direction) with initial
point (1, 0) and terminal point .x; y/,
then
cos.t/ D x and sin.t/ D y:
x
y
t
P(t) = (cos(t), sin(t))
2x 2y+ = 1
� The signs of cos.t/ and sin.t/ are determined by the quadrant in which the
terminal point of an arc t lies.
Quadrant cos.t/ sin.t/
QI positive positive
QII negative positive
QIII negative negative
QIV positive negative
� One of the most important identities in trigonometry, called the Pythagorean
Identity, is derived from the equation for the unit circle and states:
For each real number t; cos2.t/C sin2.t/ D 1:
Exercises for Section 1.2
? 1. Fill in the blanks for each of the following:
(a) For a real number t , the value of cos.t/ is defined to be the -
coordinate of the point of an arc t whose initial point
is on the whose equation is x2 Cy2 D 1.
(b) The domain of the cosine function is .
1.2. The Cosine and Sine Functions 21
(c) The maximum value of cos.t/ is and this occurs at
t D for 0 � t < 2� . The minimum value of cos.t/
is and this occurs at t D for 0 � t <
2� .
(d) The range of the cosine function is .
2. (a) For a real number t , the value of sin.t/ is defined to be the -
coordinate of the point of an arc t whose initial point
is on the whose equation is x2 Cy2 D 1.
(b) The domain of the sine function is .
(c) The maximum value of sin.t/ is and this occurs at
t D for 0 � t < 2� . The minimum value of sin.t/ is
and this occurs at t D for 0 � t <
2� .
(d) The range of the sine function is .
3. (a) Complete the following table of values.
Length of arc on Terminal point
the unit circle of the arc cos.t/ sin.t/
0 .1; 0/ 1 0
�
2
�
3�
2
2�
(b) Complete the following table of values.
Length of arc on Terminal point
the unit circle of the arc cos.t/ sin.t/
0 .1; 0/ 1 0
��
2
��
�3�
2
�2�
22 Chapter 1. The Trigonometric Functions
(c) Complete the following table of values.
Length of arc on Terminal point
the unit circle of the arc cos.t/ sin.t/
2� .1; 0/ 1 0
5�
2
3�
7�
2
4�
4. ? (a) What are the possible values of cos.t/ if it is known that sin.t/ D 3
5?
(b) What are the possible values of cos.t/ if it is known that sin.t/ D 3
5and the terminal point of t is in the second quadrant?
? (c) What is the value of sin.t/ if it is known that cos.t/ D �2
3and the
terminal point of t is in the third quadrant?
? 5. Suppose it is known that 0 < cos.t/ <1
3.
(a) By squaring the expressions in the given inequalities, what conclusions
can be made about cos2.t/?
(b) Use part (a) to write inequalities involving� cos2.t/ and then inequal-
ities involving 1 � cos2.t/.
(c) Using the Pythagorean identity, we see that sin2.t/ D 1 � cos2.t/.
Write the last inequality in part (b) in terms of sin2.t/.
(d) If we also know that sin.t/ > 0, what can we now conclude about the
value of sin.t/?
6. Use a process similar to the one in exercise (5) to complete each of the fol-
lowing:
(a) Suppose it is known that �1
4< sin.t/ < 0 and that cos.t/ > 0. What
can be concluded about cos.t/?
(b) Suppose it is known that 0 � sin.t/ � 3
7and that cos.t/ < 0. What
can be concluded about cos.t/?
1.2. The Cosine and Sine Functions 23
7. Using the four digit approximations for the cosine and sine values in Progress
Check 1.6, calculate each of the following:
� cos2.1/C sin2.1/.
� cos2.2/C sin2.2/.
� cos2.�4/C sin2.�4/.
� cos2.15/C sin2.15/.
What should be the exact value of each of these computations? Why are the
results not equal to this exact value?
24 Chapter 1. The Trigonometric Functions
1.3 Arcs, Angles, and Calculators
Focus Questions
The following questions are meant to guide our study of the material in this
section. After studying this section, we should understand the concepts mo-
tivated by these questions and be able to write precise, coherent answers to
these questions.
� How do we measure angles using degrees?
� What do we mean by the radian measure of an angle? How is the radian
measure of an angle related to the length of an arc on the unit circle?
� Why is radian measure important?
� How do we convert from radians to degrees and from degrees to radians?
� How do we use a calculator to approximate values of the cosine and sine
functions?
Introduction
The ancient civilization known as Babylonia was a cultural region based in south-
ern Mesopotamia, which is present-day Iraq. Babylonia emerged as an indepen-
dent state around 1894 BCE. The Babylonians developed a system of mathematics
that was based on a sexigesimal (base 60) number system. This was the origin of
the modern day usage of 60 minutes in an hour, 60 seconds in a minute, and 360
degrees in a circle.
Many historians now believe that for the ancient Babylonians, the year con-
sisted of 360 days, which is not a bad approximation given the crudeness of their
ancient astronomical tools. As a consequence, they divided the circle into 360
equal length arcs, which gave them a unit angle that was 1/360 of a circle or what
we now know as a degree. Even though there are 365.24 days in a year, the Babylo-
nian unit angle is still used as the basis for measuring angles in a circle. Figure 1.8
shows a circle divided up into 6 angles of 60 degrees each, which is also something
that fit nicely with the Babylonian base-60 number system.
1.3. Arcs, Angles, and Calculators 25
Figure 1.8: A circle with six 60-degree angles.
Angles
We often denote a line that is drawn through 2 points A and B by !AB . The portion
of the line !AB that starts at the point A and continues indefinitely in the direction
of the point B is called ray AB and is denoted by�!AB . The point A is the initial
point of the ray�!AB . An angle is formed by rotating a ray about its initial point.
The ray in its initial position is called the initial side of the angle, and the position
of the ray after it has been rotated is called the terminal side of the angle. The
initial point of the ray is called the vertex of the angle.
A B
C
initial side
term
inal s
ide
vertex
α
Figure 1.9: An angle including some notation.
26 Chapter 1. The Trigonometric Functions
Figure 1.9 shows the ray�!AB rotated about the point A to form an angle. The
terminal side of the angle is the ray�!AC . We often refer to this as angle BAC ,
which is abbreviated as †BAC . We can also refer to this angle as angle CAB or
†CAB . If we want to use a single letter for this angle, we often use a Greek letter
such as ˛ (alpha). We then just say the angle ˛. Other Greek letters that are often
used are ˇ (beta), (gamma), � (theta), � (phi), and � (rho).
Arcs and Angles
To define the trigonometric functions in terms of angles, we will make a simple
connection between angles and arcs by using the so-called standard position of an
angle. When the vertex of an angle is at the origin in the xy-plane and the initial
side lies along the positive x-axis, we say that the angle is in standard position.
The terminal side of the angle is then in one of the four quadrants or lies along one
of the axes. When the terminal side is in one of the four quadrants, the terminal
side determines the so-called quadrant designation of the angle. See Figure 1.10.
A B
initial side
term
inal sid
e
vertex
α
vertexvertex x
y
C
Figure 1.10: Standard position of an angle in the second quadrant.
Progress Check 1.13 (Angles in Standard Position)
Draw an angle in standard position in (1) the first quadrant; (2) the third quadrant;
and (3) the fourth quadrant.
If an angle is in standard position, then the point where the terminal side of
the angle intersects the unit circle marks the terminal point of an arc as shown in
Figure 1.11. Similarly, the terminal point of an arc on the unit circle determines
a ray through the origin and that point, which in turn defines an angle in standard
position. In this case we say that the angle is subtended by the arc. So there is
a natural correspondence between arcs on the unit circle and angles in standard
position. Because of this correspondence, we can also define the trigonometric
1.3. Arcs, Angles, and Calculators 27
functions in terms of angles as well as arcs. Before we do this, however, we need
to discuss two different ways to measure angles.
(0,0)
angle
arc
Figure 1.11: An arc and its corresponding angle.
Degrees Versus Radians
There are two ways we will measure angles – in degrees and radians. When we
measure the length of an arc, the measurement has a dimension (the length, be
it inches, centimeters, or something else). As mentioned in the introduction, the
Babylonians divided the circle into 360 regions. So one complete wrap around
a circle is 360 degrees, denoted 360ı. The unit measure of 1ı is an angle that
is 1=360 of the central angle of a circle. Figure 1.8 shows 6 angles of 60ı each.
The degree ı is a dimension, just like a length. So to compare an angle measured
in degrees to an arc measured with some kind of length, we need to connect the
dimensions. We can do that with the radian measure of an angle.
Radians will be useful in that a radian is a dimensionless measurement. We
want to connect angle measurements to arc measurements, and to do so we will
directly define an angle of 1 radian to be an angle subtended by an arc of length 1
(the length of the radius) on the unit circle as shown in Figure 1.12.
Definition. An angle of one radian is the angle in standard position on the
unit circle that is subtended by an arc of length 1 (in the positive direction).
This directly connects angles measured in radians to arcs in that we associate
a real number with both the arc and the angle. So an angle of 2 radians cuts off an
arc of length 2 on the unit circle, an angle of 3 radians cuts of an arc of length 3
on the unit circle, and so on. Figure 1.13 shows the terminal sides of angles with
measures of 0 radians, 1 radian, 2 radians, 3 radians, 4 radians, 5 radians, and 6
radians. Notice that 2� � 6:2832 and 6 < 2� as shown in Figure 1.13.
28 Chapter 1. The Trigonometric Functions
x
y
1 radian
1
Figure 1.12: One radian.
1 radian2 radians
3 radians
4 radians5 radians
6 radians
Figure 1.13: Angles with Radian
Measure 1, 2, 3, 4, 5, and 6
We can also have angles whose radian measure is negative just like we have arcs
with a negative length. The idea is simply to measure in the negative (clockwise)
direction around the unit circle. So an angle whose measure is �1 radian is the
angle in standard position on the unit circle that is subtended by an arc of length 1
in the negative (clockwise) direction.
So in general, an angle (in standard position) of t radians will correspond to an
arc of length t on the unit circle. This allows us to discuss the sine and cosine of
an angle measured in radians. That is, when we think of sin.t/ and cos.t/, we can
consider t to be:
� a real number;
� the length of an arc with initial point .1; 0/ on the unit circle;
� the radian measure of an angle in standard position.
When we draw a picture of an angle in standard position, we often draw a small
arc near the vertex from the initial side to the terminal side as shown in Figure 1.14,
which shows an angle whose measure is3
4� radians.
Progress Check 1.14 (Radian Measure of Angles)
1. Draw an angle in standard position with a radian measure of:
1.3. Arcs, Angles, and Calculators 29
x
y
Figure 1.14: An angle with measure3
4� in standard position
(a)�
2radians.
(b) � radians.
(c)3�
2radians.
(d) �3�
2radians.
2. What is the degree measure of each of the angles in part (1)?
Conversion Between Radians and Degrees
Radian measure is the preferred measure of angles in mathematics for many rea-
sons, the main one being that a radian has no dimensions. However, to effectively
use radians, we will want to be able to convert angle measurements between radi-
ans and degrees.
Recall that one wrap of the unit circle corresponds to an arc of length 2� , and
an arc of length 2� on the unit circle corresponds to an angle of 2� radians. An
angle of 360ı is also an angle that wraps once around the unit circle, so an angle
of 360ı is equivalent to an angle of 2� radians, or
� each degree is�
180radians,
� each radian is180
�degrees.
30 Chapter 1. The Trigonometric Functions
Notice that 1 radian is then180
�� 57:3ı, so a radian is quite large compared to
a degree. These relationships allow us to quickly convert between degrees and
radians. For example:
� If an angle has a degree measure of 35 degrees, then its radian measure can
be calculated as follows:
35 degrees � � radians
180 degreesD 35�
180radians:
Rewriting this fraction, we see that an angle with a measure of 35 degrees
has a radian measure of7�
36radians.
� If an angle has a radian measure of3�
10radians, then its degree measure can
be calculated as follows:
3�
10radians � 180 degrees
� radiansD 540
10degrees:
So an angle with a radian measure of3�
10has an angle measure of 54ı.
IMPORTANT NOTE: Since a degree is a dimension, we MUST include the de-
gree mark ı whenever we write the degree measure of an angle. A radian has no
dimension so there is no dimension mark to go along with it. Consequently, if we
write 2 for the measure of an angle we understand that the angle is measured in
radians. If we really mean an angle of 2 degrees, then we must write 2ı.
Progress Check 1.15 (Radian-Degree Conversions)
Complete Table 1.1 converting from degrees to radians and vice versa.
Calculators and the Trigonometric Functions
We have now seen that when we think of sin.t/ or cos.t/, we can think of t as a real
number, the length of an arc, or the radian measure of an angle. In Section 1.5, we
will see how to determine the exact values of the cosine and sine functions for a few
special arcs (or angles). For example, we will see that cos��
6
�
Dp
3
2. However,
the definition of cosine and sine as coordinates of points on the unit circle makes it
difficult to find exact values for these functions except at very special arcs. While
exact values are always best, technology plays an important role in allowing us to
1.3. Arcs, Angles, and Calculators 31
Angle in radians Angle in degrees
0 0ı
�
6�
4�
3�
290ı
120ı
3�
4135ı
150ı
180ı
Angle in radians Angle in degrees
7�
65�
44�
33�
2270ı
300ı
315ı
330ı
2� 360ı
Table 1.1: Conversions between radians and degrees.
approximate the values of the circular (or trigonometric)functions. Most hand-held
calculators, calculators in phone or tablet apps, and online calculators have a cosine
key and a sine key that you can use to approximate values of these functions, but
we must keep in mind that the calculator only provides an approximation of the
value, not the exact value (except for a small collection of arcs). In addition, most
calculators will approximate the sine and cosine of angles.
To do this, the calculator has two modes for angles: Radian and Degree. Be-
cause of the correspondence between real numbers, length of arcs, and radian mea-
sures of angles, for now, we will always put our calculators in radian mode. In
fact, we have seen that an angle measured in radians subtends an arc of that radian
measure along the unit circle. So the cosine or sine of an angle measured in
radians is the same thing as the cosine or sine of a real number when that real
number is interpreted as the length of an arc along the unit circle. (When we
study the trigonometry of triangles in Chapter 3, we will use the degree mode. For
an introductory discussion of the trigonometric functions of an angle measure in
32 Chapter 1. The Trigonometric Functions
degrees, see Exercise (4)).
Progress Check 1.16 (Using a Calculator)
In Progress Check 1.6, we used the Geogebra Applet called Terminal Points of
Arcs on the Unit Circle at http://gvsu.edu/s/JY to approximate the values
of the cosine and sine functions at certain values. For example, we found that
� cos.1/ � 0:5403,
sin.1/ � 0:8415.
� cos.2/ � �0:4161
sin.2/ � 0:9093.
� cos.�4/ � �0:6536
sin.�4/ � 0:7568.
� cos.�15/ � �0:7597
sin.�15/ � �0:6503.
Use a calculator to determine these values of the cosine and sine functions and
compare the values to the ones above. Are they the same? How are they different?
Summary of Section 1.3
In this section, we studied the following important concepts and ideas:
� An angle is formed by rotating a ray about its initial point. The ray in its
initial position is called the initial side of the angle, and the position of the
ray after it has been rotated is called the terminal side of the ray. The initial
point of the ray is called the vertex of the angle.
� When the vertex of an angle is at the origin in the xy-plane and the initial side
lies along the positive x-axis, we see that the angle is in standard position.
� There are two ways to measure angles. For degree measure, one complete
wrap around a circle is 360 degrees, denoted 360ı. The unit measure of 1ı
is an angle that is 1=360 of the central angle of a circle. An angle of one
radian is the angle in standard position on the unit circle that is subtended
by an arc of length 1 (in the positive direction).
� We convert the measure of an angle from degrees to radians by using the fact
that each degree is�
180radians. We convert the measure of an angle from
radians to degrees by using the fact that each radian is180
�degrees.
1.3. Arcs, Angles, and Calculators 33
Exercises for Section 1.3
1. Convert each of the following degree measurements for angles into radian
measures for the angles. In each case, first write the result as a fractional
multiple of � and then use a calculator to obtain a 4 decimal place approxi-
mation of the radian measure.
? (a) 15ı
? (b) 58ı(c) 112ı
(d) 210ı
? (e) �40ı
(f) �78ı
2. Convert each of the following radian measurements for angles into degree
measures for the angles. When necessary, write each result as a 4 decimal
place approximation.
? (a)3
8� radians
? (b)9
7� radians
(c) � 7
15� radians
? (d) 1 radian
(e) 2.4 radians
(f) 3 radians
3. Draw an angle in standard position of an angle whose radian measure is:
(a)1
4�
(b)1
3�
(c)2
3�
(d)5
4�
(e) �1
3�
(f) 3.4
4. In Progress Check 1.16, we used the Geogebra Applet called Terminal Points
of Arcs on the Unit Circle to approximate values of the cosine and sine func-
tions. We will now do something similar to approximate the cosine and sine
values for angles measured in degrees.
We have seen that the terminal side of an angle in standard position intersects
the unit circle in a point. We use the coordinates of this point to determine
the cosine and sine of that angle. When the angle is measured in radians, the
radian measure of the angle is the same as the arc on the unit circle subtended
by the angle. This is not true when the angle is measure in degrees, but
we can still use the intersection point to define the cosine and sine of the
angle. So if an angle in standard position has degree measurement aı, then
we define cos.aı/ to be the x-coordinate of the point of intersection of the
terminal side of that angle and the unit circle. We define sin.aı/ to be the
34 Chapter 1. The Trigonometric Functions
y-coordinate of the point of intersection of the terminal side of that angle
and the unit circle.
We will now use the Geogebra applet Angles and the Unit Circle. A web
address for this applet is
http://gvsu.edu/s/VG
For this applet, we control the value of the input aı with the slider for a.
The values of a range from �180ı to 180ı in increments of 5ı. For a given
value of aı, an angle in standard position is drawn and the coordinates of the
point of intersection of the terminal side of that angle and the unit circle are
displayed. Use this applet to approximate values for each of the following:
? (a) cos .10ı/ and sin .10ı/.
(b) cos .60ı/ and sin .60ı/.
(c) cos .135ı/ and sin .135ı/.
? (d) cos .�10ı/ and sin .�10ı/.
(e) cos .�135ı/ and sin .�135ı/.
(f) cos .85ı/ and sin .85ı/.
5. Exercise (4) must be completed before doing this exercise. Put the calculator
you are using in Degree mode. Then use the calculator to determine approx-
imate values of the cosine and sine functions in Exercise (4). Are the values
the same? How are they different?
1.4. Velocity and Angular Velocity 35
1.4 Velocity and Angular Velocity
Focus Questions
The following questions are meant to guide our study of the material in this
section. After studying this section, we should understand the concepts mo-
tivated by these questions and be able to write precise, coherent answers to
these questions.
� What is arc length?
� What is the difference between linear velocity and angular velocity?
� What are the formulas that relate linear velocity to angular velocity?
Beginning Activity
1. What is the formula for the circumference C of a circle whose radius is r?
2. Suppose person A walks along the circumference of a circle with a radius
of 10 feet, and person B walks along the circumference of a circle of radius
20 feet. Also, suppose it takes both A and B 1 minute to walk one-quarter
of the circumference of their respective circles (one-quarter of a complete
revolution). Who walked the most distance in one minute?
3. Suppose both people continue walking at the same pace they did for the first
minute. How many complete revolutions of the circle will each person walk
in 8 minutes? In 10 minutes?
Arc Length on a Circle
In Section 1.3, we learned that the radian measure of an angle was equal to the
length of the arc on the unit circle associated with that angle. So an arc of length
1 on the unit circle subtends an angle of 1 radian. There will be times when it will
also be useful to know the length of arcs on other circles that subtend the same
angle.
In Figure 1.15, the inner circle has a radius of 1, the outer circle has a radius of
r , and the angle shown has a measure of � radians. So the arc length on the unit
36 Chapter 1. The Trigonometric Functions
x
y
r1
θ
θ
s
Figure 1.15: Arcs subtended by an angle of 1 radian.
circle subtended by the angle is � , and we have used s to represent the arc length
on the circle of radius r subtended by the angle.
Recall that the circumference of a circle of radius r is 2�r while the circum-
ference of the circle of radius 1 is 2� . Therefore, the ratio of an arc length s on the
circle of radius r that subtends an angle of � radians to the corresponding arc on
the unit circle is2�r
2�D r . So it follows that
s
�D 2�r
2�D r
s D r�
Definition. On a circle of radius r , the arc length s intercepted by a central
angle with radian measure � is
s D r�:
Note: It is important to remember that to calculate arc length2 , we must measure
the central angle in radians.
2It is not clear why the letter s is usually used to represent arc length. One explanation is that the
arc “subtends” an angle.
1.4. Velocity and Angular Velocity 37
Progress Check 1.17 (Using the Formula for Arc Length)
Using the circles in the beginning activity for this section:
1. Use the formula for arc length to determine the arc length on a circle of
radius 10 feet that subtends a central angle of�
2radians. Is the result equal
to one-quarter of the circumference of the circle?
2. Use the formula for arc length to determine the arc length on a circle of
radius 20 feet that subtends a central angle of�
2radians. Is the result equal
to one-quarter of the circumference of the circle?
3. Determine the arc length on a circle of radius 3 feet that is subtended by an
angle of 22ı.
Why Radians?
Degree measure is familiar and convenient, so why do we introduce the concept of
radian measure for angles? This is a good question, but one with a subtle answer.
As we just saw, the length s of an arc on a circle of radius r subtended by angle
of � radians is given by s D r� , so � D s
r. As a result, a radian measure is a
ratio of two lengths (the quotient of the length of an arc by a radius of a circle),
which makes radian measure a dimensionless quantity. Thus, a measurement in
radians can just be thought of as a real number. This is convenient for dealing
with arc length (and angular velocity as we will soon see), and it will also be useful
when we study periodic phenomena in Chapter 2. For this reason radian measure is
universally used in mathematics, physics, and engineering as opposed to degrees,
because when we use degree measure we always have to take the degree dimension
into account in computations. This means that radian measure is actually more nat-
ural from a mathematical standpoint than degree measure.
Linear and Angular Velocity
The connection between an arc on a circle and the angle it subtends measured
in radians allows us to define quantities related to motion on a circle. Objects
traveling along circular paths exhibit two types of velocity: linear and angular
velocity. Think of spinning on a merry-go-round. If you drop a pebble off the
edge of a moving merry-go-round, the pebble will not drop straight down. Instead,
it will continue to move forward with the velocity the merry-go-round had the
38 Chapter 1. The Trigonometric Functions
moment the pebble was released. This is the linear velocity of the pebble. The
linear velocity measures how the arc length changes over time.
Consider a point P moving at a constant linear velocity along the circumfer-
ence of a circle of radius r . This is called uniform circular motion. Suppose that
P moves a distance of s units in time t . The linear velocity v of the point P is the
distance it traveled divided by the time elapsed. That is, v D s
t. The distance s is
the arc length and we know that s D r� .
Definition. Consider a point P moving at a constant linear velocity along the
circumference of a circle of radius r . The linear velocity v of the point P is
given by
v D s
tD r�
t;
where � , measured in radians, is the central angle subtended by the arc of
length s.
Another way to measure how fast an object is moving at a constant speed on a
circular path is called angular velocity. Whereas the linear velocity measures how
the arc length changes over time, the angular velocity is a measure of how fast the
central angle is changing over time.
Definition. Consider a point P moving with constant velocity along the cir-
cumference of a circle of radius r on an arc that corresponds to a central angle
of measure � (in radians). The angular velocity ! of the point is the radian
measure of the angle � divided by the time t it takes to sweep out this angle.
That is
! D �
t:
Note: The symbol ! is the lower case Greek letter “omega.” Also, notice that the
angular velocity does not depend on the radius r .
This is a somewhat specialized definition of angular velocity that is slightly
different than a common term used to describe how fast a point is revolving around
a circle. This term is revolutions per minute or rpm. Sometimes the unit revolu-
tions per second is used. A better way to represent revolutions per minute is to use
the “unit fraction”rev
min. Since 1 revolution is 2� radians, we see that if an object
1.4. Velocity and Angular Velocity 39
is moving at x revolutions per minute, then
! D xrev
min� 2�rad
revD x.2�/
rad
min:
Progress Check 1.18 (Determining Linear Velocity)
Suppose a circular disk is rotating at a rate of 40 revolutions per minute. We wish
to determine the linear velocity v (in feet per second) of a point that is 3 feet from
the center of the disk.
1. Determine the angular velocity ! of the point in radians per minute. Hint:
Use the formula
! D xrev
min� 2�rad
rev:
2. We now know ! D �
t. So use the formula v D r�
tto determine v in feet
per minute.
3. Finally, convert the linear velocity v in feet per minute to feet per second.
Notice that in Progress Check 1.18, once we determined the angular velocity,
we were able to determine the linear velocity. What we did in this specific case we
can do in general. There is a simple formula that directly relates linear velocity to
angular velocity. Our formula for linear velocity is v D s
tD r�
t. Notice that we
can write this is v D r�
t. That is, v D r!.
Consider a point P moving with constant (linear) velocity v along the circum-
ference of a circle of radius r . If the angular velocity is !, then
v D r!:
So in Progress Check 1.18, once we determined that ! D 80�rad
min, we could
determine v as follows:
v D r! D .3 ft/
�
80�rad
min
�
D 240�ft
min:
Notice that since radians are “unit-less”, we can drop them when dealing with
equations such as the preceding one.
40 Chapter 1. The Trigonometric Functions
Example 1.19 (Linear and Angular Velocity)
The LP (long play) or 331
3rpm vinyl record is an analog sound storage medium
and has been used for a long time to listen to music. An LP is usually 12 inches
or 10 inches in diameter. In order to work with our formulas for linear and angular
velocity, we need to know the angular velocity in radians per time unit. To do this,
we will convert 331
3revolutions per minute to radians per minute. We will use the
fact that 331
3D 100
3.
! D 100
3
rev
min� 2� rad
1 rev
D 200�
3
rad
min
We can now use the formula v D r! to determine the linear velocity of a point on
the edge of a 12 inch LP. The radius is 6 inches and so
v D r!
D .6 inches/
�
200�
3
rad
min
�
D 400�inches
min
It might be more convenient to express this as a decimal value in inches per second.
So we get
v D 400�inches
min� 1 min
60 sec
� 20:944inches
sec
The linear velocity is approximately 20.944 inches per second.
Progress Check 1.20 (Linear and Angular Velocity)
For these problems, we will assume that the Earth is a sphere with a radius of 3959
miles. As the Earth rotates on its axis, a person standing on the Earth will travel in
a circle that is perpendicular to the axis.
1. The Earth rotates on its axis once every 24 hours. Determine the angular
velocity of the Earth in radians per hour. (Leave your answer in terms of the
number � .)
1.4. Velocity and Angular Velocity 41
2. As the Earth rotates, a person standing on the equator will travel in a circle
whose radius is 3959 miles. Determine the linear velocity of this person in
miles per hour.
3. As the Earth rotates, a person standing at a point whose latitude is 60ı north
will travel in a circle of radius 2800 miles. Determine the linear velocity of
this person in miles per hour and feet per second.
Summary of Section 1.4
In this section, we studied the following important concepts and ideas:
� On a circle of radius r , the arc length s intercepted by a central angle with
radian measure � is
s D r�:
� Uniform circular motion is when a point moves at a constant linear velocity
along the circumference of a circle. The linear velocity is the arc length
traveled by the point divided by the time elapsed. Whereas the linear velocity
measures how the arc length changes over time, the angular velocity is a
measure of how fast the central angle is changing over time. The angular
velocity of the point is the radian measure of the angle divided by the time it
takes to sweep out this angle.
� For a point P moving with constant (linear) velocity v along the circumfer-
ence of a circle of radius r , we have
v D r!;
where ! is the angular velocity of the point.
Exercises for Section 1.4
? 1. Determine the arc length (to the nearest hundredth of a unit when necessary)
for each of the following.
(a) An arc on a circle of radius 6 feet that is intercepted by a central angle
of2�
3radians. Compare this to one-third of the circumference of the
circle.
42 Chapter 1. The Trigonometric Functions
(b) An arc on a circle of radius 100 miles that is intercepted by a central
angle of 2 radians.
(c) An arc on a circle of radius 20 meters that is intercepted by a central
angle of13�
10radians.
(d) An arc on a circle of radius 10 feet that is intercepted by a central angle
of 152 degrees.
2. In each of the following, when it is possible, determine the exact measure of
the central angle in radians. Otherwise, round to the nearest hundredth of a
radian.
? (a) The central angle that intercepts an arc of length 3� feet on a circle of
radius 5 feet.
? (b) The central angle that intercepts an arc of length 18 feet on a circle of
radius 5 feet.
(c) The central angle that intercepts an arc of length 20 meters on a circle
of radius 12 meters.
3. In each of the following, when it is possible, determine the exact measure of
central the angle in degrees. Otherwise, round to the nearest hundredth of a
degree.
? (a) The central angle that intercepts an arc of length 3� feet on a circle of
radius 5 feet.
? (b) The central angle that intercepts an arc of length 18 feet on a circle of
radius 5 feet.
(c) The central angle that intercepts an arc of length 20 meters on a circle
of radius 12 meters.
(d) The central angle that intercepts an arc of length 5 inches on a circle of
radius 5 inches.
(e) The central angle that intercepts an arc of length 12 inches on a circle
of radius 5 inches.
4. Determine the distance (in miles) that the planet Mars travels in one week in
its path around the sun. For this problem, assume that Mars completes one
complete revolution around the sun in 687 days and that the path of Mars
around the sun is a circle with a radius of 227.5 million miles.
1.4. Velocity and Angular Velocity 43
? 5. Determine the distance (in miles) that the Earth travels in one day in its path
around the sun. For this problem, assume that Earth completes one complete
revolution around the sun in 365.25 days and that the path of Earth around
the sun is a circle with a radius of 92.96 million miles.
6. A compact disc (CD) has a diameter of 12 centimeters (cm). Suppose that
the CD is in a CD-player and is rotating at 225 revolutions per minute. What
is the angular velocity of the CD (in radians per second) and what is the
linear velocity of a point on the edge of the CD?
7. A person is riding on a Ferris wheel that takes 28 seconds to make a complete
revolution. Her seat is 25 feet from the axle of the wheel.
(a) What is her angular velocity in revolutions per minute? Radians per
minute? Degrees per minute?
(b) What is her linear velocity?
(c) Which of the quantities angular velocity and linear velocity change if
the person’s seat was 20 feet from the axle instead of 25 feet? Compute
the new value for any value that changes. Explain why each value
changes or does not change.
8. A small pulley with a radius of 3 inches is connected by a belt to a larger
pulley with a radius of 7.5 inches (See Figure 1.16). The smaller pulley is
connected to a motor that causes it to rotate counterclockwise at a rate of 120
rpm (revolutions per minute). Because the two pulleys are connected by the
belt, the larger pulley also rotates in the counterclockwise direction.
Figure 1.16: Two Pulleys Connected by a Belt
(a) Determine the angular velocity of the smaller pulley in radians per
minute.
44 Chapter 1. The Trigonometric Functions
? (b) Determine the linear velocity of the rim of the smaller pulley in inches
per minute.
(c) What is the linear velocity of the rim of the larger pulley? Explain.
(d) Find the angular velocity of the larger pulley in radians per minute.
(e) How many revolutions per minute is the larger pulley turning?
9. A small pulley with a radius of 10 centimeters inches is connected by a belt to
a larger pulley with a radius of 24 centimeters inches (See Figure 1.16). The
larger pulley is connected to a motor that causes it to rotate counterclockwise
at a rate of 75 rpm (revolutions per minute). Because the two pulleys are
connected by the belt, the smaller pulley also rotates in the counterclockwise
direction.
(a) Determine the angular velocity of the larger pulley in radians per minute.
? (b) Determine the linear velocity of the rim of the large pulley in inches
per minute.
(c) What is the linear velocity of the rim of the smaller pulley? Explain.
(d) Find the angular velocity of the smaller pulley in radians per second.
(e) How many revolutions per minute is the smaller pulley turning?
10. The radius of a car wheel is 15 inches. If the car is traveling 60 miles per
hour, what is the angular velocity of the wheel in radians per minute? How
fast is the wheel spinning in revolutions per minute?
11. The mean distance from Earth to the moon is 238,857 miles. Assuming the
orbit of the moon about Earth is a circle with a radius of 238,857 miles and
that the moon makes one revolution about Earth every 27.3 days, determine
the linear velocity of the moon in miles per hour. Research the distance of
the moon to Earth and explain why the computations that were just made are
approximations.
1.5. Common Arcs and Reference Arcs 45
1.5 Common Arcs and Reference Arcs
Focus Questions
The following questions are meant to guide our study of the material in this
section. After studying this section, we should understand the concepts mo-
tivated by these questions and be able to write precise, coherent answers to
these questions.
� How do we determine the values for cosine and sine for arcs whose
endpoints are on the x-axis or the y-axis?
� What are the exact values of cosine and sine for t D �
6, t D �
4, and
t D �
3?
� What is the reference arc for a given arc? How do we determine the
reference arc for a given arc?
� How do we use reference arcs to calculate the values of the cosine and
sine at other arcs that have�
6,
�
4, or
�
3as reference arcs?
Beginning Activity
Figure 1.17 shows a unit circle with the terminal points for some arcs between 0
and 2� . In addition, there are four line segments drawn on the diagram that form
a rectangle. The line segments go from: (1) the terminal point for t D �
6to the
terminal point for t D 5�
6; (2) the terminal point for t D 5�
6to the terminal point
for t D 7�
6; (3) the terminal point for t D 7�
6to the terminal point for t D 11�
6;
and (4) the terminal point for t D 11�
6to the terminal point for t D �
6.
1. What are the approximate values of cos��
6
�
and sin��
6
�
?
2. What are the approximate values of cos
�
5�
6
�
and sin
�
5�
6
�
?
46 Chapter 1. The Trigonometric Functions
π
6
π
4
3
π
3
5
π
4
5
π
6
7 π
6
11
π
4
π
3
π
2π
3
2
π
4
7
π
3
4
π
6
5
π
2
3
0
Figure 1.17: Some Arcs on the unit circle
3. What are the approximate values of cos
�
7�
6
�
and sin
�
7�
6
�
?
4. What are the approximate values of cos
�
11�
6
�
and sin
�
11�
6
�
?
5. Draw a similar rectangle on Figure 1.17 connecting the terminal points for
t D �
4, t D 3�
4, t D 5�
4, and t D 7�
4. How do the cosine and sine values
for these arcs appear to be related?
Our task in this section is to determine the exact cosine and sine values for all
of the arcs whose terminal points are shown in Figure 1.17. We first notice that we
already know the cosine and sine values for the arcs whose terminal points are on
one of the coordinate axes. These values are shown in the following table.
t 0�
2�
3�
22�
cos.t/ 1 0 �1 0 1
sin.t/ 0 1 0 �1 0
Table 1.2: Cosine and Sine Values
1.5. Common Arcs and Reference Arcs 47
The purpose of the beginning activity was to show that we determine the values
of cosine and sine for the other arcs by finding only the cosine and sine values for
the arcs whose terminal points are in the first quadrant. So this is our first task. To
do this, we will rely on some facts about certain right triangles. The three triangles
we will use are shown in Figure 1.18.
Figure 1.18: Special Right Triangles.
In each figure, the hypotenuse of the right triangle has a length of c units.
The lengths of the other sides are determined using the Pythagorean Theorem. An
explanation of how these lengths were determined can be found on page 425 in
Appendix C. The usual convention is to use degree measure for angles when we
work with triangles, but we can easily convert these degree measures to radian
measures.
� A 30ı angle has a radian measure of�
6radians.
� A 45ı angle has a radian measure of�
4radians.
� A 60ı angle has a radian measure of�
3radians.
The Values of Cosine and Sine at t D�
6Figure 1.19 shows the unit circle in the first quadrant with an arc in standard posi-
tion of length�
6. The terminal point of the arc is the point P and its coordinates
are�
cos��
6
�
; sin��
6
��
. So from the diagram, we see that
x D cos��
6
�
and y D sin��
6
�
:
48 Chapter 1. The Trigonometric Functions
P
Q
π
6π/6
1
x
y
Figure 1.19: The arc�
6and its associated angle.
As shown in the diagram, we form a right triangle by drawing a line from P
that is perpendicular to the x-axis and intersects the x-axis at Q. So in this right
triangle, the angle associated with the arc is�
6radians or 30ı. From what we
know about this type of right triangle, the other acute angle in the right triangle is
60ı or�
3radians. We can then use the results shown in the triangle on the left in
Figure 1.18 to conclude that x Dp
3
2and y D 1
2. (Since in this case, c D 1.)
Therefore, we have just proved that
cos��
6
�
Dp
3
2and sin
��
6
�
D 1
2:
Progress Check 1.21 (Comparison to the Beginning Activity)
In the beginning activity for this section, we used the unit circle to approximate
the values of the cosine and sine functions at t D �
6, t D 5�
6, t D 7�
6, and
t D 11�
6. We also saw that these values are all related and that once we have
values for the cosine and sine functions at t D �
6, we can use our knowledge of
the four quadrants to determine these function values at t D 5�
6, t D 7�
6, and
1.5. Common Arcs and Reference Arcs 49
t D 11�
6. Now that we know that
cos��
6
�
Dp
3
2and sin
��
6
�
D 1
2;
determine the exact values of each of the following:
1. cos
�
5�
6
�
and sin
�
5�
6
�
.
2. cos
�
7�
6
�
and sin
�
7�
6
�
.
3. cos
�
11�
6
�
and sin
�
11�
6
�
.
The Values of Cosine and Sine at t D�
4Figure 1.20 shows the unit circle in the first quadrant with an arc in standard posi-
tion of length�
4. The terminal point of the arc is the point P and its coordinates
are�
cos��
4
�
; sin��
4
��
. So from the diagram, we see that
x D cos��
4
�
and y D sin��
4
�
:
π
4π
4
P
π
4
x
y
Q
1
Figure 1.20: The arc�
4and its associated angle.
As shown in the diagram, we form a right triangle by drawing a line from P
that is perpendicular to the x-axis and intersects the x-axis at Q. So in this right
50 Chapter 1. The Trigonometric Functions
triangle, the acute angles are�
4radians or 45ı. We can then use the results shown
in the triangle in the middle of Figure 1.18 to conclude that x Dp
2
2and y D
p2
2.
(Since in this case, c D 1.) Therefore, we have just proved that
cos��
4
�
Dp
2
2and sin
��
4
�
Dp
2
2:
Progress Check 1.22 (Comparison to the Beginning Activity)
Now that we know that
cos��
4
�
Dp
2
2and sin
��
4
�
Dp
2
2;
use a method similar to the one used in Progress Check 1.21 to determine the exact
values of each of the following:
1. cos
�
3�
4
�
and sin
�
3�
4
�
.
2. cos
�
5�
4
�
and sin
�
5�
4
�
.
3. cos
�
7�
4
�
and sin
�
7�
4
�
.
The Values of Cosine and Sine at t D�
3Figure 1.21 shows the unit circle in the first quadrant with an arc in standard posi-
tion of length�
3. The terminal point of the arc is the point P and its coordinates
are�
cos��
3
�
; sin��
3
��
. So from the diagram, we see that
x D cos��
3
�
and y D sin��
3
�
:
As shown in the diagram, we form a right triangle by drawing a line from P
that is perpendicular to the x-axis and intersects the x-axis at Q. So in this right
triangle, the angle associated with the arc is�
3radians or 60ı. From what we know
about this type of right triangle, the other acute angle in the right triangle is 30ı
or�
6radians. We can then use the results shown in the triangle on the right in
1.5. Common Arcs and Reference Arcs 51
P
Qx
1
y
π
3
π
3
Figure 1.21: The arc�
3and its associated angle.
Figure 1.18 to conclude that x D 1
2and y D
p3
2. (Since in this case, c D 1.)
Therefore, we have just proved that
cos��
3
�
D 1
2and sin
��
3
�
Dp
3
2:
Reference Arcs (Reference Angles)
In the beginning activity for this section and in Progress Checks 1.21 and 1.22, we
saw that we could relate the coordinates of the terminal point of an arc of length
greater than�
2on the unit circle to the coordinates of the terminal point of an arc
of length between 0 and�
2on the unit circle. This was intended to show that we
can do this for any angle of length greater than�
2, and this means that if we know
the values of the cosine and sine for any arc (or angle) between 0 and�
2, then we
can find the values of the cosine and sine for any arc at all. The arc between 0 and�
2to which we relate a given arc of length greater than
�
2is called a reference arc.
Definition. The reference arc Ot (read t -hat) for an arc t is the smallest non-
negative arc (always considered non-negative) between the terminal point of
the arc t and the closer of the two x-intercepts of the unit circle. Note that the
two x-intercepts of the unit circle are .�1; 0/ and .1; 0/.
52 Chapter 1. The Trigonometric Functions
The concept of reference arc is illustrated in Figure 1.22. Each of the thicker
arcs has length Ot and it can be seen that the coordinates of the points in the second,
third, and fourth quadrants are all related to the coordinates of the point in the first
quadrant. The signs of the coordinates are all determined by the quadrant in which
the point lies.
(cos( ), sin( ))(- cos( ), sin( ))
(cos( ), - sin( ))
t
t
t
t
t
t
tt
(- cos( ), - sin( ))tt tt
Figure 1.22: Reference arcs.
How we calculate a reference arc for a given arc of length t depends upon the
quadrant in which the terminal point of t lies. The diagrams in Figure 1.23 on
page 54 illustrate how to calculate the reference arc for an arc of length t with
0 � t � 2� .
In Figure 1.23, we see that for an arc of length t with 0 � t � 2� :
� If�
2< t < � , then the point intersecting the unit circle and the x axis that
is closest to the terminal point of t is .�1; 0/. So the reference arc is � � t .
In this case, Figure 1.23 shows that
1.5. Common Arcs and Reference Arcs 53
cos.� � t / D � cos.t/ and sin.� � t / D sin.t/:
� If � < t <3�
2, then the point intersecting the unit circle and the x axis that
is closest to the terminal point of t is .�1; 0/. So the reference arc is t � � .
In this case, Figure 1.23 shows that
cos.t � �/ D � cos.t/ and sin.t � �/ D � sin.t/:
� If3�
2< t < 2� , then the point intersecting the unit circle and the x axis that
is closest to the terminal point of t is .1; 0/. So the reference arc is 2� � t .
In this case, Figure 1.23 shows that
cos.2� � t / D cos.t/ and sin.2� � t / D � sin.t/:
Progress Check 1.23 (Reference Arcs – Part 1)
For each of the following arcs, draw a picture of the arc on the unit circle. Then
determine the reference arc for that arc and draw the reference arc in the first quad-
rant.
1. t D 5�
42. t D 4�
53. t D 5�
3
Progress Check 1.24 (Reference Arcs – Part 2)
Although we did not use the term then, in Progress Checks 1.21 and 1.22, we used
the facts that t D �
6and t D �
4were the reference arcs for other arcs to determine
the exact values of the cosine and sine functions for those other arcs. Now use the
values of cos��
3
�
and sin��
3
�
to determine the exact values of the cosine and
sine functions for each of the following arcs:
54 Chapter 1. The Trigonometric Functions
(cos(t), sin(t))
x
y
t
If the arc t is in Quadrant I, then t
is its own reference arc.
(cos(t), sin(t))
x
y
(cos(π - t), sin(π - t))
π - tπ - t
If the arc t is in Quadrant II, then
� � t is its reference arc.
x
y
(cos(t), sin(t))
(cos(t - π), sin(t - π))
t - π
t - π
If the arc t is in Quadrant III, then
t � � is its reference arc.
(cos(t), sin(t))
x
y
(cos(2π - t), sin(2π - t))
2π - t
2π - t
If the arc t is in Quadrant IV, then
2� � t is its reference arc.
Figure 1.23: Reference arcs
1. t D 2�
32. t D 4�
33. t D 5�
3
Reference Arcs for Negative Arcs
Up to now, we have only discussed reference arcs for positive arcs, but the same
principles apply when we use negative arcs. Whether the arc t is positive or neg-
ative, the reference arc for t is the smallest non-negative arc formed by the termi-
nal point of t and the nearest x-intercept of the unit circle. For example, the arc
t D ��
4is in the fourth quadrant, and the closer of the two x-intercepts of the unit
circle is .1; 0/. So the reference arc is Ot D �
4as shown in Figure 1.24.
1.5. Common Arcs and Reference Arcs 55
π
4
π
4
_
A
B
(1, 0)
Figure 1.24: Reference Arc for t D �
4.
Since we know that the point A has coordinates
p2
2;
p2
2
!
, we conclude that
the point B has coordinates
p2
2;�p
2
2
!
, and so
cos�
��
4
�
Dp
2
2and sin
�
��
4
�
D �p
2
2:
Progress Check 1.25 (Reference Arcs for Negative Arcs)
For each of the following arcs, determine the reference arc and the exact values of
the cosine and sine functions.
1. t D ��
62. t D �2�
33. t D �5�
4
Example 1.26 (Using Reference Arcs)
Sometimes we can use the concept of a reference arc even if we do not know the
length of the arc but do know the value of the cosine or sine function. For example,
suppose we know that
0 < t <�
2and sin.t/ D 2
3:
56 Chapter 1. The Trigonometric Functions
Are there any conclusions we can make with this information? Following are some
possibilities.
1. We can use the Pythagorean identity to determine cos.t/ as follows:
cos2.t/C sin2.t/ D 1
cos2.t/ D 1 ��
2
3
�2
cos2.t/ D 5
9
Since t is in the first quadrant, we know that cos.t/ is positive, and hence
cos.t/ Dr
5
9Dp
5
3:
2. Since 0 < t <�
2, t is in the first quadrant. Hence, � � t is in the second
quadrant and the reference arc is t . In the second quadrant we know that the
sine is positive, so we can conclude that
sin.� � t / D sin.t/ D 2
3:
Progress Check 1.27 (Working with Reference Arcs)
Following is information from Example 1.26:
0 < t <�
2and sin.t/ D 2
3:
Use this information to determine the exact values of each of the following:
1. cos.� � t /
2. sin.� C t /
3. cos.� C t /
4. sin.2� � t /
Summary of Section 1.5
In this section, we studied the following important concepts and ideas:
� The values of cos.t/ and sin.t/ for arcs whose terminal points are on one of
the coordinate axes are shown in Table 1.3 below.
1.5. Common Arcs and Reference Arcs 57
� Exact values for the cosine and sine functions at�
6,
�
4, and
�
3are known
and are shown in Table 1.3 below.
� A reference arc for an arc t is the arc (always considered nonnegative) be-
tween the terminal point of the arc t and point intersecting the unit circle and
the x-axis closest to it.
� If t is an arc that has an arc Ot as a reference arc, then j cos.t/j and j cos.Ot /j are
the same. Whether cos.t/ D cos.Ot / or cos.t/ D � cos.Ot/ is determined by
the quadrant in which the terminal side of t lies. The same is true for sin.t/.
� We can determine the exact values of the cosine and sine functions at any arc
with�
6,
�
4, or
�
3as reference arc. These arcs between 0 and 2� are shown
in Figure 1.17. The results are summarized in Table 1.3 below.
t x D cos.t/ y D sin.t/
0 1 0
�
6
p3
2
1
2
�
4
p2
2
p2
2
�
3
1
2
p3
2�
20 1
2�
3�1
2
p3
2
3�
4�p
2
2
p2
2
5�
6�p
3
2
1
2
t x D cos.t/ y D sin.t/
� �1 0
7�
6�p
3
2�1
2
5�
4�p
2
2�p
2
2
4�
3�1
2�p
3
2
3�
20 �1
5�
3
1
2�p
3
2
7�
4
p2
2�p
2
2
11�
6
p3
2�1
2
Table 1.3: Exact values of the cosine and sine functions.
58 Chapter 1. The Trigonometric Functions
Exercises for Section 1.5
? 1. A unit circle is shown in each of the following with information about an arc
t . In each case, use the information on the unit circle to determine the values
of t , cos.t/, and sin.t/.
(a)
1
2, ?( )
x
y
t
(b)
(?, 1)y
t
x
(c)
y
t
x
(d)
y
tx
2. Determine the exact value for each of the following expressions and then use
a calculator to check the result. For example,
cos.0/C sin��
3
�
D 1Cp
3
2� 1:8660:
? (a) cos2��
6
�
? (b) 2 sin2��
4
�
C cos.�/
(c)cos
��
6
�
sin��
6
�
(d) 3 sin��
2
�
C cos��
4
�
3. For each of the following, determine the reference arc for the given arc and
draw the arc and its reference arc on the unit circle.
? (a) t D 4�
3
? (b) t D 13�
8
(c) t D 9�
4
? (d) t D �4�
3
(e) t D �7�
5
(f) t D 5
1.5. Common Arcs and Reference Arcs 59
4. For each of the following, draw the given arc t on the unit circle, determine
the reference arc for t , and then determine the exact values for cos.t/ and
sin.t/.
? (a) t D 5�
6
(b) t D 5�
4
(c) t D 5�
3
? (d) t D �2�
3
(e) t D �7�
4
(f) t D 19�
6
5. (a) Use a calculator (in radian mode) to determine five-digit approxima-
tions for cos.4/ and sin.4/.
(b) Use a calculator (in radian mode) to determine five-digit approxima-
tions for cos.4 � �/ and sin.4 � �/.
(c) Use the concept of reference arcs to explain the results in parts (a) and
(b).
6. Suppose that we have the following information about the arc t .
0 < t <�
2and sin.t/ D 1
5:
Use this information to determine the exact values of each of the following:
? (a) cos.t/
(b) sin.� � t /
(c) cos.� � t /
? (d) sin.� C t /
(e) cos.� C t /
(f) sin.2� � t /
7. Suppose that we have the following information about the arc t .
�
2< t < � and cos.t/ D �2
3:
Use this information to determine the exact values of each of the following:
(a) sin.t/
(b) sin.� � t /
(c) cos.� � t /
(d) sin.� C t /
(e) cos.� C t /
(f) sin.2� � t /
60 Chapter 1. The Trigonometric Functions
8. Make sure your calculator is in Radian Mode.
(a) Use a calculator to find an eight-digit approximation of sin��
6C �
4
�
D
sin
�
5�
12
�
.
(b) Determine the exact value of sin��
6
�
C sin��
4
�
.
(c) Use a calculator to find an eight-digit approximation of your result in
part (b). Compare this to your result in part (a). Does it seem that
sin��
6C �
4
�
D sin��
6
�
C sin��
4
�
‹
(d) Determine the exact value of sin��
6
�
cos��
4
�
C cos��
6
�
sin��
4
�
.
(e) Determine an eight-digit approximation of your result in part (d).
(f) Compare the results in parts (a) and (e). Does it seem that
sin��
6C �
4
�
D sin��
6
�
cos��
4
�
C cos��
6
�
sin��
4
�
‹
9. This exercise provides an alternate
method for determining the exact values
of cos��
4
�
and sin��
4
�
. The diagram
to the right shows the terminal point
P.x; y/ for an arc of length t D �
4on
the unit circle. The points A.1; 0/ and
B.0; 1/ are also shown.
Since the point B is the terminal point of
the arc of length�
2, we can conclude that
the length of the arc from P to B is also�
4. Because of this, we conclude that the
point P lies on the line y D x as shown
in the diagram. Use this fact to determine
the values of x and y. Explain why this
proves that
cos��
4
�
Dp
2
2and sin
��
4
�
Dp
2
2:
1.5. Common Arcs and Reference Arcs 61
10. This exercise provides an alternate
method for determining the exact values
of cos��
6
�
and sin��
6
�
. The diagram
to the right shows the terminal point
P.x; y/ for an arc of length t D �
6on the unit circle. The points A.1; 0/,
B.0; 1/, and C.x;�y/ are also shown.
Notice that B is the terminal point of the
arc t D �
2, and C is the terminal point of
the arc t D ��
6.
We now notice that the length of the arc from P to B is
�
2� �
6D �
3:
In addition, the length of the arc from C to P is
�
6� ��
6D �
3:
This means that the distance from P to B is equal to the distance from C to
P .
(a) Use the distance formula to write a formula (in terms of x and y) for
the distance from P to B .
(b) Use the distance formula to write a formula (in terms of x and y) for
the distance from C to P .
(c) Set the distances from (a) and (b) equal to each other and solve the
resulting equation for y. To do this, begin by squaring both sides of the
equation. In order to solve for y, it may be necessary to use the fact
that x2 C y2 D 1.
(d) Use the value for y in (c) and the fact that x2 C y2 D 1 to determine
the value for x.
Explain why this proves that
cos��
6
�
Dp
3
2and sin
��
6
�
D 1
2:
62 Chapter 1. The Trigonometric Functions
11. This exercise provides an alternate
method for determining the exact values
of cos��
3
�
and sin��
3
�
. The diagram
to the right shows the terminal point
P.x; y/ for an arc of length t D �
3on the unit circle. The points A.1; 0/,
B.0; 1/, and S
p3
2;1
2
!
are also shown.
Notice that B is the terminal point of the
arc t D �
2.
From Exercise (10), we know that S is
the terminal point of an arc of length�
6.
We now notice that the length of the arc from A to P is�
3. In addition, since
the length of the arc from A to B is�
2and the and the length of the arc from
A to P is�
3, the length of the arc from P to B is equal
�
2� �
3D �
6:
Since both of the arcs have length�
6, the distance from A to S is equal to
the distance from P to B .
(a) Use the distance formula to determine the distance from A to S .
(b) Use the distance formula to write a formula (in terms of x and y) for
the distance from P to B .
(c) Set the distances from (a) and (b) equal to each other and solve the
resulting equation for y. To do this, begin by squaring both sides of the
equation. In order to solve for y, it may be necessary to use the fact
that x2 C y2 D 1.
(d) Use the value for y in (c) and the fact that x2 C y2 D 1 to determine
the value for x.
Explain why this proves that
cos��
3
�
D 1
2and sin
��
3
�
Dp
3
2:
1.6. Other Trigonometric Functions 63
1.6 Other Trigonometric Functions
Focus Questions
The following questions are meant to guide our study of the material in this
section. After studying this section, we should understand the concepts mo-
tivated by these questions and be able to write precise, coherent answers to
these questions.
� How is the tangent function defined? What is the domain of the tangent
function?
� What are the reciprocal functions and how are they defined? What are
the domains of each of the reciprocal functions?
We defined the cosine and sine functions as the coordinates of the terminal
points of arcs on the unit circle. As we will see later, the sine and cosine give
relations for certain sides and angles of right triangles. It will be useful to be able
to relate different sides and angles in right triangles, and we need other circular
functions to do that. We obtain these other circular functions – tangent, cotangent,
secant, and cosecant – by combining the cosine and sine together in various ways.
Beginning Activity
Using radian measure:
1. For what values of t is cos.t/ D 0?
2. For what values of t is sin.t/ D 0?
3. In what quadrants is cos.t/ > 0? In what quadrants is sin.t/ > 0?
4. In what quadrants is cos.t/ < 0? In what quadrants is sin.t/ < 0?
The Tangent Function
Next to the cosine and sine, the most useful circular function is the tangent.3
3The word tangent was introduced by Thomas Fincke (1561-1656) in his Flenspurgensis Geome-
triae rotundi libri XIIII where he used the word tangens in Latin. From “Earliest Known Uses of
Some of the Words of Mathematics at http://jeff560.tripod.com/mathword.html.
64 Chapter 1. The Trigonometric Functions
Definition. The tangent function is the quotient of the sine function divided
by the cosine function. So the tangent of a real number t is defined to besin.t/
cos.t/for those values t for which cos.t/ ¤ 0. The common abbreviation for
the tangent of t is
tan.t/ D sin.t/
cos.t/:
In this definition, we need the restriction that cos.t/ ¤ 0 to make sure the
quotient is defined. Since cos.t/ D 0 whenever t D �
2C k� for some integer k,
we see that tan.t/ is defined when t ¤ �
2C k� for all integers k. So
The domain of the tangent function is the set of all real numbers t for
which t ¤ �
2C k� for every integer k.
Notice that although the domain of the sine and cosine functions is all real numbers,
this is not true for the tangent function.
When we worked with the unit circle definitions of cosine and sine, we often
used the following diagram to indicate signs of cos.t/ and sin.t/ when the terminal
point of the arc t is in a given quadrant.
x
y
sin (t) > 0 sin (t) > 0
cos (t) > 0
cos (t) > 0cos (t) < 0
cos (t) < 0
sin (t) < 0 sin (t) < 0
Progress Check 1.28 (Signs and Values of the Tangent Function)
Considering t to be an arc on the unit circle, for the terminal point of t :
1. In which quadrants is tan.t/ positive?
2. In which quadrants is tan.t/ negative?
1.6. Other Trigonometric Functions 65
3. For what values of t is tan.t/ D 0?
4. Complete Table 1.4, which gives the values of cosine, sine, and tangent at
the common reference arcs in Quadrant I.
t cos.t/ sin.t/ tan.t/
0 0
�
6
1
2
�
4
p2
2
�
3
p3
2
�
21
Table 1.4: Values of the Tangent Function
Just as with the cosine and sine, if we know the values of the tangent function
at the reference arcs, we can find its values at any arc related to a reference arc. For
example, the reference arc for the arc t D 5�
3is
�
3. So
tan
�
5�
3
�
Dsin
�
5�
3
�
cos
�
5�
3
�
D� sin
��
3
�
cos��
3
�
D�p
3
21
2
D �p
3
66 Chapter 1. The Trigonometric Functions
We can shorten this process by just using the fact that tan��
3
�
Dp
3 and that
tan
�
5�
3
�
< 0 since the terminal point of the arc5�
3is in the fourth quadrant.
tan
�
5�
3
�
D � tan��
3
�
D �p
3:
Progress Check 1.29 (Values of the Tangent Function)
1. Determine the exact values of tan
�
5�
4
�
and tan
�
5�
6
�
.
2. Determine the exact values of cos.t/ and tan.t/ if it is known that sin.t/ D 1
3and tan.t/ < 0.
The Reciprocal Functions
The remaining circular or trigonometric functions are reciprocals of the cosine,
sine, and tangent functions. Since these functions are reciprocals, their domains
will be all real numbers for which the denominator is not equal to zero. The first
we will introduce is the secant function. 4 function.
Definition. The secant function is the reciprocal of the cosine function. So
the secant of a real number t is defined to be1
cos.t/for those values t where
cos.t/ ¤ 0. The common abbreviation for the secant of t is
sec.t/ D 1
cos.t/:
Since the tangent function and the secant function use cos.t/ in a denominator,
they have the same domain. So
4The term secant was introduced by was by Thomas Fincke (1561-1656) in his Thomae Finkii
Flenspurgensis Geometriae rotundi libri XIIII, Basileae: Per Sebastianum Henricpetri, 1583. Vieta
(1593) did not approve of the term secant, believing it could be confused with the geometry term.
He used Transsinuosa instead. From “Earliest Known Uses of Some of the Words of Mathematics at
http://jeff560.tripod.com/mathword.html.
1.6. Other Trigonometric Functions 67
The domain of the secant function is the set of all real numbers t for
which t ¤ �
2C k� for every integer k.
Next up is the cosecant function. 5
Definition. The cosecant function is the reciprocal of the sine function. So
the cosecant of a real number t is defined to be1
sin.t/for those values t where
sin.t/ ¤ 0. The common abbreviation for the cosecant of t is
csc.t/ D 1
sin.t/:
Since sin.t/ D 0 whenever t D k� for some integer k, we see that
The domain of the cosecant function is the set of all real numbers t for
which t ¤ k� for every integer k.
Finally, we have the cotangent function. 6
Definition. The cotangent function is the reciprocal of the tangent function.
So the cotangent of a real number t is defined to be1
tan.t/for those values t
where tan.t/ ¤ 0. The common abbreviation for the cotangent of t is
cot.t/ D 1
tan.t/:
Since tan.t/ D 0 whenever t D k� for some integer k, we see that
The domain of the cotangent function is the set of all real numbers t for
which t ¤ k� for every integer k.
5Georg Joachim von Lauchen Rheticus appears to be the first to use the term cosecant (as cosecans
in Latin) in his Opus Palatinum de triangulis. From Earliest Known Uses of Some of the Words of
Mathematics at http://jeff560.tripod.com/mathword.html.6The word cotangent was introduced by Edmund Gunter in Canon Triangulorum (Table of Arti-
ficial Sines and Tangents) where he used the term cotangens in Latin. From Earliest Known Uses of
Some of the Words of Mathematics at http://jeff560.tripod.com/mathword.html.
68 Chapter 1. The Trigonometric Functions
A Note about Calculators
When it is not possible to determine exact values of a trigonometric function, we
use a calculator to determine approximate values. However, please keep in mind
that most calculators only have keys for the sine, cosine, and tangent functions.
With these calculators, we must use the definitions of cosecant, secant, and cotan-
gent to determine approximate values for these functions.
Progress Check 1.30 (Values of Trigonometric Functions
When possible, find the exact value of each of the following functional values.
When this is not possible, use a calculator to find a decimal approximation to four
decimal places.
1. sec
�
7�
4
�
2. csc���
4
�
3. tan
�
7�
8
�
4. cot
�
4�
3
�
5. csc.5/
Progress Check 1.31 (Working with Trigonometric Functions
1. If cos.x/ D 1
3and sin.x/ < 0, determine the exact values of sin.x/, tan.x/,
csc .x/, and cot.x/.
2. If sin.x/ D � 7
10and tan.x/ > 0, determine the exact values of cos.x/ and
cot.x/.
3. What is another way to write .tan.x//.cos.x//?
Summary of Section 1.6
In this section, we studied the following important concepts and ideas:
� The tangent function is the quotient of the sine function divided by the
cosine function. That is,
tan.t/ D sin.t/
cos.t/;
for those values t for which cos.t/ ¤ 0. The domain of the tangent func-
tion is the set of all real numbers t for which t ¤ �
2C k� for every integer
k.
1.6. Other Trigonometric Functions 69
� The reciprocal functions are the secant, cosecant, and tangent functions.
Reciprocal Function Domain
sec.t/ D 1
cos.t/The set of real numbers t for which t ¤ �
2C k�
for every integer k.
csc.t/ D 1
sin.t/The set of real numbers t for which t ¤ k� for
every integer k.
cot.t/ D 1
tan.t/The set of real numbers t for which t ¤ k� for
every integer k.
Exercises for Section 1.6
? 1. Complete the following table with the exact values of each functional value
if it is defined.t cot.t/ sec.t/ csc.t/
0
�
6�
4�
3�
2
2. Complete the following table with the exact values of each functional value
if it is defined.
t cot.t/ sec.t/ csc.t/2�
37�
67�
4
��
3
�
70 Chapter 1. The Trigonometric Functions
3. Determine the quadrant in which the terminal point of each arc lies based on
the given information.
? (a) cos.x/ > 0 and tan.x/ < 0.
? (b) tan.x/ > 0 and csc.x/ < 0.
(c) cot.x/ > 0 and sec.x/ > 0.
(d) sin.x/ < 0 and sec.x/ > 0.
(e) sec.x/ < 0 and csc.x/ > 0.
(f) sin.x/ < 0 and cot.x/ > 0.
? 4. If sin.t/ D 1
3and cos.t/ < 0, determine the exact values of cos.t/, tan.t/,
csc.t/, sec.t/, and cot.t/.
5. If cos.t/ D �3
5and sin.t/ < 0, determine the exact values of sin.t/, tan.t/,
csc.t/, sec.t/, and cot.t/.
6. If sin.t/ D �2
5and tan.t/ < 0, determine the exact values of cos.t/, tan.t/,
csc.t/, sec.t/, and cot.t/.
7. If sin.t/ D 0:273 and cos.t/ < 0, determine the three-digit approximations
for cos.t/, tan.t/, csc.t/, sec.t/, and cot.t/.
8. In each case, determine the arc t that satisfies the given conditions or explain
why no such arc exists.
? (a) tan.t/ D 1, cos.t/ D � 1p2
, and 0 < t < 2� .
? (b) sin.t/ D 1, sec.t/ is undefined, and 0 < t < � .
(c) sin.t/ Dp
2
2, sec.t/ D �
p2, and 0 < t < � .
(d) sec.t/ D � 2p3
, tan.t/ Dp
3, and 0 < t < 2� .
(e) csc.t/ Dp
2, tan.t/ D �1, and 0 < t < 2� .
9. Use a calculator to determine four-digit decimal approximations for each of
the following.
(a) csc.1/
(b) tan
�
12�
5
�
(c) cot.5/
(d) sec
�
13�
8
�
(e) sin2.5:5/
(f) 1C tan2.2/
(g) sec2.2/
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